Abstract view
Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups


Yanchang Han,
School of Mathematical Sciences, South China Normal University , Guangzhou, 510631, China
Yongsheng Han,
Department of Mathematics, Auburn University , Auburn, AL 36849, USA
Ji Li,
Department of Mathematics, Macquarie University , Sydney NSW 2109, Australia
Chaoqiang Tan,
Department of Mathematics, Shantou University , Shantou, Guangdong 515041, China
Abstract
The Marcinkiewicz multipliers are $L^{p}$ bounded for $1\lt p\lt \infty
$ on the
Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$
(MÃ¼ller, Ricci and Stein). This is
surprising in the sense that these multipliers are invariant
under a two parameter
group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while
there is
\emph{no} two parameter group of \emph{automorphic} dilations
on $\mathbb{H}
^{n}$. The purpose of this paper is to establish a theory of
the flag Lipschitz space on the Heisenberg group
$\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the
sense
`intermediate' between the classical Lipschitz space on the Heisenberg
group
$\mathbb{H}^{n}$ and the product Lipschitz space on
$\mathbb{C}^{n}\times \mathbb{R}$. We characterize this flag
Lipschitz space
via the LittlewoodPaley theory and prove
that flag singular integral operators, which include the
Marcinkiewicz multipliers, are bounded on these flag Lipschitz
spaces.
Keywords: 
Heisenberg group, Marcinkiewicz multiplier, Flag singular integral, Flag Lipschitz space, reproducing formula, Discrete LittlewoodPaley analysis
Heisenberg group, Marcinkiewicz multiplier, Flag singular integral, Flag Lipschitz space, reproducing formula, Discrete LittlewoodPaley analysis
